A norm ratio inequality - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T19:08:01Z http://mathoverflow.net/feeds/question/57228 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57228/a-norm-ratio-inequality A norm ratio inequality Aryeh Kontorovich 2011-03-03T09:13:31Z 2011-03-03T09:13:31Z <p>Let $y,z\in(0,1)^n$ satisfy $||y||_1 = ||z||_1=1$.</p> <p>Then $$\frac{||z||_3}{||z||_2} \le K_n ||z/y||_\infty \frac{||y||_3}{||y||_2}$$ where $z/y\in\mathbb R^n$ is the coordinate-wise quotient of $z$ and $y$ and $K_n>0$ is some constant depending only on $n$. What's the best $K_n$ that makes this true? What if $2,3$ are replaced by general $p,q$?</p>